A 2+1-Dimensional Non-Isothermal Magnetogasdynamic System. Hamiltonian-Ermakov Integrable Reduction

A 2+1-dimensional anisentropic magnetogasdynamic system with a polytropic gas law is shown to admit an integrable elliptic vortex reduction when $\gamma=2$ to a nonlinear dynamical subsystem with underlying integrable Hamiltonian-Ermakov structure. Exact solutions of the magnetogasdynamic system are thereby obtained which describe a rotating elliptic plasma cylinder. The semi-axes of the elliptical cross-section, remarkably, satisfy a Ermakov-Ray-Reid system.


Introduction
Neukirch et al. [6,7,8] have investigated 2+1-dimensional magnetogasdynamic systems via a solution approach in which the nonlinear acceleration terms in the Lundquist momentum equation either vanish or are conservative. By contrast, in recent work [12,20] an elliptic vortex ansatz was adopted in 2+1-dimensional isothermal magnetogasdynamics and underlying integrable Ermakov-Ray-Reid structure was isolated. In [20], magnetogasdynamic pulsrodontype solutions were constructed analogous to those originally derived in elliptic warm-core theory in [13]. The pulsrodons describe an elliptical plasma cylinder bounded by a vacuum. The timedependent semi-axes of the elliptical cross-section of the cylinder were shown to be governed by an integrable Hamiltonian Ermakov system.
The present work concerns an extension of that of [12] to a non-isothermal rotating magnetogasdynamic version of a spinning non-conducting gas cloud system with origin in work of Ovsiannikov [9] and Dyson [2]. A nonlinear dynamical subsystem is derived which is again remarkably, shown to have integrable Hamiltonian Ermakov-Ray-Reid structure. Moreover, a Lax pair for the dynamical system is constructed.

4)
∂S ∂t + q · ∇S = 0, (2.5) where the velocity q and magnetic field H are given by respectively, while the gas law adopts the polytropic form with p = ρT. (2.8) In the above, the magneto-gas density ρ(x, t), pressure p(x, t), entropy S(x, t), temperature T (x, t) and magnetic flux A(x, t) are all assumed to be dependent only on x = xi + yj and time t. In addition, f is the Coriolis constant, µ the magnetic permeability and h(x, t) the transverse component of the magnetic field. Insertion of the representation (2.6) into Faraday's law (2.4) produces the convective constraint ∂A ∂t + q · ∇A = 0 (2.9) together with ∂h ∂t + div(hq) = 0, which holds automatically if we set Here, a novel two-parameter (m, n) pressure-density ansatz is introduced. In the magnetogasdynamic study of [7], a relation p ∼ ρ was adopted, while pressure-density relations of the type p ∼ ρ 2 arise in astrophysical contexts [21]. A parabolic pressure-density law was recently employed in 2+1-dimensional isothermal magnetogasdynamics in [20] and pulsrodon-type solutions were isolated.
In the present non-isothermal context, substitution of (2.10) in (2.8) produces the temperature distribution while the entropy distribution adopts the form The energy equation now requires that whence, on use of the continuity equation (2.1) On substitution of (2.6) and (2.10) into the momentum equation (2.2), it is seen that Attention is here restricted to the separable case whence, substitution into (2.9) and use of the continuity equation yieldṡ Here, we proceed with where n is the parameter involving in the relation (2.11), so that div q = 1 n Ψ Ψ (2.14) and A = ρ n Ψ(t).
Hence, as in the case of the spinning non-conducting gas cloud analysis of Ovsiannikov [9] and Dyson [2], the divergence of the velocity is dependent only on time. Moreover, the relation (2.13) shows that and it is observed that this condition holds identically with where α i (i = 0, 1, 2) are arbitrary constants of integration. In addition, the isentropic condition (2.5) together with the polytropic gas law (2.7) and the continuity equation (2.1) show that div q = 1 1 − γṪ T whence, on use of (2.11), It is seen that in view of (2.14), these relations are indeed consistent with (2.15)-(2.17).
In summary, the magnetogasdynamic system now reduces to consideration of the nonlinear coupled system ∂ρ ∂t + div(ρq) = 0, where m = 1, together with the additional conditions (2.15)-(2.17). The inherent nonlinearity of the system (2.18) remains a major impediment to analytic progress. It is noted also that the system (2.18) 1,3 is overdetermined since it is implicitly constrained by the requirement (2.18) 2 that div q be a function of t only.

An elliptic vortex ansatz. A dynamical system reduction
Here, integrable nonlinear dynamical subsystems of the magnetogasdynamic system (2.18) are sought via an elliptic vortex ansatz of the type Insertion of (3.1) into the continuity equation yields If we now proceed with and then it is seen that (2.18) 3 reduces to The relation (3.7) implies thatε 0 = 0 whence (2.15) shows that the adiabatic index γ = 2, while (3.8) and (2.16) together require augmented by the auxiliary linear equations It is noted that the relation (2.18) 2 together with (3.6) shows thaṫ While ρ 0 is given in terms of Ψ via (3.5). The constraints (3.7) and (3.8) are to be adjoined and their admissibility will be examined subsequently.
In what follows, it proves convenient to proceed in terms of new variables, namely These quantities were originally introduced in a hydrodynamic context (see e.g. [13]). Therein, G and G R correspond, in turn, to the divergence and spin of the velocity field, while G S and G N represent shear and normal deformation rates. The system (3.3) and (3.4) together with (3.10) now reduces to the nonlinear dynamical systeṁ It is observed that the introduction of the pressure-density parameters (m, n) and ε 2 leads to a generalisation of the nonlinear dynamical systems obtained in [12,13,14,20].
If we now introduce the quantity Ω via G = 2Ω Ω then (3.12) 1 and (3.12) 8 show, in turn, that and While the relation (3.13) yields where c 0 , c I and ν denote arbitrary constants of integration. Two conditions which are key to the subsequent development and which may be established by appeal to the original system (3.12) are now recorded. These represent extensions of results obtained in a hydrodynamic context [13,14].
New Ω-modulated variables involving the pressure-density parameter m are now introduced according tō augmented by the relations (3.14) and (3.15) together with a nonlinear equation for Ω, namely The reduced dynamical system (3.18) together with (3.19)  In the former case, by virtue of (3.16), the magnetic flux A vanishes so that the magnetic field H is purely transverse and the dynamical system (3.18) and (3.19) is not thereby constrained. Here, we proceed with the latter case, so that the system (3.18) and (3.19) is additionally constrained by the requirement Ω 2B = const and (3.18) 1 yields Finally, for ε 2 , the relations (2.17), (3.6) and (3.16) combine to show that which it subsequently proves convenient to re-write in this form

Integrals of motion and parametrisation
Under the constraint (3.20), the nonlinear dynamical system (3.18) is readily shown to admit the key integrals of motion where c II , c III , c IV and c V are constants of integration. The relations (4.1) and (4.2) may be conveniently parametrised according tō G S = − c III +B sin θ(t),Ḡ N = + c III +B cos θ(t).
while conditions (3.18) 2,3 reduce to a single relation, namely while elimination of θ − φ in (4.7) and (4.8) respectively shows thaṫ It remains to consider the nonlinear equation (3.19) for Ω, namely which, by virtue of (3.20), reduces to a generalisation of the classical Steen-Ermakov equation [3,22], namelÿ Further, on use of Theorem 1, it may be readily shown that there is the necessary requirement (cf. [13])( This holds automatically here with Elimination of θ − φ between (4.6) and (4.9) now yieldṡ whence on use of (4.13) The latter equation is required to be compatible with the 1 st integral of (4.12), namelẏ αδ Ω 4 + k = 0 and these are indeed seen to be consistent subject to the relations In summary, a multi-parameter class of exact vortex solutions of the original 2+1-dimensional magnetogasdynamic system has been generated with the velocity components u 1 , u 2 , v 1 , v 2 and the quantities a, b, c, ρ 0 in the density representation given, in turn, by where the angles φ and θ are obtained by integration of (4.10) and (4.11), respectively while Ω is given by an elliptic integral resulting from (4.14).
The magnetic flux A is given by while the temperature T and entropy distribution S are determined by (2.11) and (2.12), respectively.

Hamiltonian Ermakov structure
The nonlinear dynamical system (3.12) may be shown to have remarkable underlying structure in that it will be seen to reduce to consideration of an integrable Ermakov-Ray-Reid type system Such systems have their origin in the work of Ermakov [3] and were introduced by Ray and Reid in [10,11]. Extension to 2+1-dimensions were presented in [15] and to multi-component systems in [18]. The main theoretical interest in the system resides in its admittance of a distinctive integral of motion, namely, the Ray-Reid invariant Ermakov-Ray-Reid systems arise, in particular, in a variety of contexts in nonlinear optics (see e.g. [16,17] and references cited therein).
Here, we proceed withp(t) =q(t) = 0 in the ansatz (3.1), since the translation termsp(t),q(t) are readily re-introduced by use of a Lie group invariance of the magnetogasdynamic system.
The semi-axes of the time-modulated ellipse a(t)x 2 + 2b(t)xy + cy 2 + h 0 (t) = 0, ac − b 2 > 0, are now given by where it is required that It is readily established that the semi-axes Φ, Ψ are governed by a Ermakov-Ray-Reid system, namelÿ where and Ω is given in terms of the ratio of the semi-axes via the relation In addition, the Ermakov-Ray-Reid system (5.1) is seen to be Hamiltonian with invariant and accordingly, is amenable to the general procedure described in detail in [14]. It is remarkable indeed that the semi-axes Φ and Ψ of the time modulated ellipse associated with the density representation in (3.1), are governed by an integrable Ermakov-Ray-Reid system, albeit of some complexity. In fact, a Ermakov-Ray-Reid system may also be associated with the velocity components, at least, in a particular reduction. Attention is here restricted, as in the work of Dyson [2] on non-conducting gas clouds, to irrotational motions in the absence of a Coriolis term.
Thus, here we set in (3.2) corresponding to the subclass of exact solutions in (4.15) with θ = 0, φ = π/2 anḋ The continuity equation, via (3.3), yieldṡ Moreover, (3.4) shows that In the above, c I , c II , c III and c * III are arbitrary non-zero constants of integration. The momentum equation gives Insertion of the expressions (5.2) into (5.3) gives whence, in view of the relation (3.20), we again obtain a Ermakov-Ray-Reid system, namelÿ with the Ray-Reid invariant It is observed moreover, that the system (5.4) is also Hamiltonian with additional integral of motion

A Lax pair formulation
It is now shown, following a procedure analogous to that set down in the spinning gas cloud analysis of [16], that the nonlinear dynamical system (3.3) and (3.10) admits an associated Lax pair representation. In this connection, it is seen that the nonlinear dynamical system given by (3.3) together with (3.10) arising from the ansatz (3.1) and (3.2) may be written in the compact matrix form aṡ where L, E are given by (3.2) and Moreover, the relations (3.4) and (3.11) yielḋ A gauge transformation is now introduced viã whereL * =L − 1 2 (trL)I denotes the trace-free part ofL. Moreover, the trace-free part of (6.5) yieldṡ while its trace gives where L(λ) =L * + λP, M(λ) =Q + λL * + λ 2 P.
An analogous result has been obtained in the case of non-conducting rotating gas clouds in [19]. As in that work, there is an interesting Steen-Ermakov connection. Thus, on setting Σ = Ω −1 then the relation (6.8) is readily shown to reduce to a Steen-Ermakov equation, namely Σ + (detL * − trĒ)Σ = f 2 4Σ 3 .
Results of [19] related to the Lax pair for a spinning gas cloud system carry over mutatis mutandis to the Lax pair (6.11) obtained in the present magnetogasdynamic study. Thus, the linear system (6.11) is gauge equivalent to the standard Lax pair for the stationary reduction of the integrable cubic nonlinear Schrödinger equation. The connection may be made in the manner set down in [19].

Conclusion
It has been shown via an elliptic vortex ansatz that there is hidden integrable structure of Ermakov-Ray-Reid type underlying a 2+1-dimensional non-isothermal magnetogasdynamic system. The Ermakov variables turn out to have a natural physical interpretation as the semi-axes of the time-modulated density representation. Moreover, a Lax pair for the original nonlinear dynamical subsystem has been constructed. The preceeding and previous studies such as that in [4] suggest that a general investigation of the occurrence of integrable Ermakov-Ray-Reid structure in 2+1-dimensional hydrodynamic systems would be of interest. It is noted that Hamiltonian-Ermakov type systems have been additionally investigated in [1,5].